"Distinguished varieties" are a special class of algebraic
curves in C^2 that exit the bidisk through the distinguished boundary
(aka the torus). We shall discuss connections with the polynomials
that define these curves and polynomials with no zeros on the bidisk,
and use a powerful "sums of squares" formula (actually a two variable
version of the Christoffel-Darboux formula for orthogonal polynomials)
to a prove a determinantal representation of distinguished varieties.
As an application of our approach, we will prove a certain bounded
analytic "extension" theorem.